Theorem bnj1071 35527

Description: Technical lemma for bnj69 35560. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 35319	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7869	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6367	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 19	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896  ∅c0 4279  {csn 4584   E cep 5547   Fr wfr 5598  Ord word 6351  ωcom 7861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-ss 3916  df-uni 4868  df-tr 5213  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6355  df-on 6356  df-om 7862

This theorem is used by:  bnj1030  35537  bnj1133  35539